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Is complexity a source of incompleteness?

Authors :
Calude, Cristian S.
Jürgensen, Helmut
Source :
Advances in Applied Mathematics. Jul2005, Vol. 35 Issue 1, p1-15. 15p.
Publication Year :
2005

Abstract

Abstract: In this paper we prove Chaitin''s “heuristic principle,”the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself, for an appropriate measure of complexity. We show that the measure is invariant under the change of the Gödel numbering. For this measure, the theorems of a finitely-specified, sound, consistent theory strong enough to formalize arithmetic which is arithmetically sound (like Zermelo–Fraenkel set theory with choice or Peano Arithmetic) have bounded complexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. Previous results showing that incompleteness is not accidental, but ubiquitous are here reinforced in probabilistic terms: the probability that a true sentence of length n is provable in the theory tends to zero when n tends to infinity, while the probability that a sentence of length n is true is strictly positive. [Copyright &y& Elsevier]

Details

Language :
English
ISSN :
01968858
Volume :
35
Issue :
1
Database :
Academic Search Index
Journal :
Advances in Applied Mathematics
Publication Type :
Academic Journal
Accession number :
17855472
Full Text :
https://doi.org/10.1016/j.aam.2004.10.003